Bayesian games with a continuum of states

We show that every Bayesian game with purely atomic types has a measurable Bayesian equilibrium when the common knowl- edge relation is smooth. Conversely, for any common knowledge rela- tion that is not smooth, there exists a type space that yields this common knowledge relation and payoffs such that the resulting Bayesian game will not have any Bayesian equilibrium. We show that our smoothness condition also rules out two paradoxes involving Bayesian games with a continuum of types: the impossibility of having a common prior on components when a common prior over the entire state space exists, and the possibility of interim betting/trade even when no such trade can be supported ex ante

Projections and functions of Nash equilibria

We show that any non-empty compact semi-algebraic subset of mixed action profiles on a fixed player set can be represented as the projection of the set of equilibria of a game in which additional binary players have been added. Even stronger, we show that any semi-algebraic continuous function, or even any semi-algebraic upper-semicontinuous correspondence with non-empty convex values, from a bounded semi-algebraic set to the unit cube can be represented as the projection of an equilibrium correspondence of a game with binary players in which payoffs depend on parameters from the domain of the function or correspondence in a multi-affine way. Some extensions are also presented

Measurable selection for purely atomic games

A general selection theorem is presented constructing a measurable mapping from a state space to a parameter space under the assumption that the state space can be decomposed as a collection of countable equivalence classes under a smooth equivalence relation. It is then shown how this selection theorem can be used as a general purpose tool for proving the existence of measurable equilibria in broad classes of several branches of games when an appropriate smoothness condition holds, including Bayesian games with atomic knowledge spaces, stochastic games with countable orbits, and graphical games of countable degree—examples of a subclass of games with uncountable state spaces that we term purely atomic games. Applications to repeated games with symmetric incomplete information and acceptable bets are also presented

Corrigendum to “Discounted stochastic games with no stationary Nash equilibrium: two examples”

Levy (2013) presented examples of discounted stochastic games that do not have stationary equilibria. The second named author has pointed out that one of these examples is incorrect. In addition to describing the details of this error, this note presents a new example by the first named author that succeeds in demonstrating that discounted stochastic games with absolutely continuous transitions can fail to have stationary equilibria

Determinacy of games with Stochastic Eventual Perfect Monitoring

We consider an infinite two-player stochastic zero-sum game with a Borel winning set, in which the opponent's actions are monitored via stochastic private signals. We introduce two conditions of the signalling structure: Stochastic Eventual Perfect Monitoring (SEPM) and Weak Stochastic Eventual Perfect Monitoring (WSEPM). When signals are deterministic these two conditions coincide and by a recent result due to Shmaya (2011) entail determinacy of the game. We generalize Shmaya's (2011) result and show that in the stochastic learning environment SEPM implies determinacy while WSEPM does not

An update on continuous-time stochastic games of fixed duration

This paper shows that continuous-time stochastic games of fixed duration need not possess equilibria in Markov strategies. The example requires payoffs and transitions to depend on time in a continuous but irregular (almost nowhere almost differentiable) way. This example offers a correction to the erroneous construction presented previously in Levy (Dyn Games Appl 3(2):279–312, 2013. https://doi.org/10.1007/s13235-012-0067-2)

On games without approximate equilibria

This note shows that the work by Simon and Tomkowicz (Israel J Math 227(1):215–231, 2018) answers another outstanding open question in game theory in addition to the non-existence of approximate Harsányi equilibrium in Bayesian games: it shows that strategic form games with bounded and separately continuous payoffs need not possess approximate equilibria

Dense orbits of the Bayesian updating group action

We study dynamic properties of the group action on the simplex that is induced by Bayesian updating. We show that, generically, the orbits are dense in the simplex, although one must make use of the entire group, hence departing from straightforward Bayesian updating. We demonstrate also the necessity of the genericity of the signalling structure, a relationship to descriptive set theoretical concepts, and applications thereof to repeated games of incomplete information, as well a strengthening concerning the group action on itself

Uniformly supported approximate equilibria in families of games

No abstract available